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A000162 Number of 3-dimensional polyominoes (or polycubes) with n cells.
(Formerly M1845 N0731)
38
1, 1, 2, 8, 29, 166, 1023, 6922, 48311, 346543, 2522522, 18598427, 138462649, 1039496297, 7859514470, 59795121480, 457409613979, 3516009200564, 27144143923583, 210375361379518, 1636229771639924, 12766882202755783 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,3
COMMENTS
Here two polycubes that differ by reflection are considered different. - Joerg Arndt, Apr 26 2023
Number of oriented polyominoes with n cubical cells of the regular tiling with Schläfli symbol {4,3,4}. For oriented polyominoes, chiral pairs are counted as two. - Robert A. Russell, Mar 21 2024
REFERENCES
C. J. Bouwkamp, personal communication.
J. R. Long and R. H. Holm, Enumeration and structural classification of clusters derived from parent solids ..., J. Amer. Chem. Soc., 116 (1984), 9987-10002.
W. F. Lunnon, Symmetry of cubical and general polyominoes, pp. 101-108 of R. C. Read, editor, Graph Theory and Computing. Academic Press, NY, 1972.
W. F. Lunnon, personal communication.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Nina Bohlmann and Ralf Benölken, Complex Tasks: Potentials and Pitfalls, Mathematics (2020) Vol. 8, No. 10, 1780.
C. J. Bouwkamp & N. J. A. Sloane, Correspondence, 1971.
A. Clarke, Polycubes.
A. Clarke, The 8 tetracubes.
Kevin L. Gong, Polyominoes Home Page.
M. Keller, Counting polyforms.
David A. Klarner, Some results concerning polyominoes, Fibonacci Quarterly 3 (1965), 9-20.
Eric Weisstein's World of Mathematics, Polycube.
FORMULA
a(n) = 2*A038119 - A007743.
a(n) = A000105 + A006759.
a(n) = A038119(n) + A371397(n) = 2*A371397(n) + A007743(n) - Robert A. Russell, Mar 21 2024
EXAMPLE
Table showing total number and numbers with each group order.
-------------------------------------------------------------
The last 7 columns form sequences A066453, A066454, A066273, A066281, A066283, A066287, A066288.
.n ...A000162 ..group:.1.....2...3...4.6.8.24
.1 .........1..........0.....0...0...0.0.0..1
.2 .........1..........0.....0...0...0.0.1..0
.3 .........2..........0.....1...0...0.0.1..0
.4 .........8..........1.....4...1...0.0.2..0
.5 ........29.........17....10...0...0.0.2..0
.6 .......166........127....34...0...3.1.1..0
.7 ......1023........941....71...4...5.0.1..1
.8 ......6922.......6662...246...0..11.0.2..1
.9 .....48311......47771...522...3..11.0.4..0
10 ....346543.....344708..1783..24..24.2.2..0
11 ...2522522....2518713..3765...4..35.0.5..0
12 ..18598427...18585455.12858..18..84.5.7..0
13 .138462649..138434899.27496.151..92.2.8..1
14 1039496297.1039401564.94525..25.174.4.5..0
CROSSREFS
Cf. A038119 (unoriented), A371397 (chiral), A007743 (achiral), A001931 (fixed).
Sequence in context: A185033 A287045 A009419 * A052437 A131318 A010749
KEYWORD
nonn,nice,hard,more,changed
AUTHOR
EXTENSIONS
The old value for a(11), 2522572, was corrected by Achim Flammenkamp to 2522522, Feb 15 1999.
a(13)-a(14) from Brendan Owen (brendan_owen(AT)yahoo.com), Dec 27, 2001
a(15)-a(16) from Herman Jamke (hermanjamke(AT)fastmail.fm), May 05 2007
a(17)-a(20) from Stanley Dodds, Dec 11 2023
a(21)-a(22) (using Dodds's algorithm) from Phillip Thompson, Feb 07 2024
STATUS
approved

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Last modified March 29 11:45 EDT 2024. Contains 371278 sequences. (Running on oeis4.)